Algebraic space

In mathematics, an algebraic space is a generalization of the affine schemes of algebraic geometry introduced by Michael Artin for use in deformation theory.

=Definition=

An algebraic space X is comprised of an affine scheme U and a closed subscheme R ⊂ U × U satisfying the following two conditions:

:1. R is an equivalence relation as a subset of U × U :2. The projections p i : R → U onto each factor are étale morphisms.

If a third condition

:3. R is the trivial equivalence relation over each connected component of U

is satisfied, then the algebraic space will be a scheme in the usual sense. Thus, an algebraic space allows a single connected component of U to covering space X with finitely many sheets. The point set underlying the algebraic space X is then given by | U | / | R | as a set of equivalence classes.

Let Y be an algebraic space defined by an equivalence relation S ⊂ V × V . The set Hom( Y , X ) of morphisms of algebraic spaces is then defined by the condition that it makes the descent (category theory)

:mathrm{Hom}(Y, X) ightarrow mathrm{Hom}(V, X) { ightarrowatop ightarrow} mathrm{Hom}(S, X)

exact (this definition is motivated by a descent theorem of Grothendieck for surjective étale maps of affine schemes). With these definitions, the algebraic spaces form a category (mathematics).

Let U be an affine scheme over a field k defined by a system of polynomials g ( x ), x = x 1, …, x n , let k { x 1, …, x n } denote the ring (mathematics) of algebraic functions in x over k , and let X ⊂ U × U be an algebraic space.

The appropriate stalks Ã? X , x on X are then defined to be the local rings of algebraic functions defined by Ã? U , u , where u ∈ U is a point lying over x and Ã? U , u is the local ring corresponding to u of the ring k { x 1, …, x n } / ( g ) of algebraic functions on U .

A point on an algebraic space is said to be smooth if Ã? X , x ≅ k { z 1, …, z d } for some indeterminate (variable)s z 1, …, z d . The dimension of X at x is then just defined to be d .

A morphism f : Y → X of algebraic spaces is said to be étale at y ∈ Y (where x = f ( y )) if the induced map on stalks Ã? X , x → Ã? Y , y is an isomorphism.

The structure sheaf O X on the algebraic space X is defined by associating the ring of functions O ( V ) on V (defined by étale maps from V to the affine line A1 in the sense just defined) to any algebraic space V which is étale over X .

=Facts about algebraic spaces=

Algebraic curves are schemes.

Non-singular algebraic surfaces are schemes.

Algebraic spaces with group structure are schemes.

Not every singular algebraic surface is a scheme.

Not every non-singular 3-dimensional algebraic space is a scheme.

Every algebraic space contains a dense open affine subscheme, and the complement of such a subscheme always has Codimension ≥ 1. Thus algebraic spaces are in a sense close to affine schemes.

=Applications=

To be written

=Reference=

Artin, Michael. Algebraic Spaces . Yale University Press, 1971.